1. Field of the Invention
The invention relates to numerical methods for determining properties of semiconductor devices by simulation, and more particularly to the use of such methods where the body to be modeled has anisotropic properties.
2. Description of Related Art
Numerical modeling and simulation has become an indispensable tool for the optimal design of semiconductor devices. As the dimensions of semiconductor devices continue to decrease, numerical simulation plays an increasingly important role in helping transistor designers develop strategies to overcome the design limits. Numerical modeling is used extensively also for simulating fabrication processes, such as the diffusion of dopants in a substrate. In digital and analog circuit design and analysis, numerical modeling of circuit simulations is used to assist in circuit verification. In device analysis, numerical modeling is used to determine parameters for compact models of such devices that can be used to analyze circuits that include such devices. For example, numerical modeling can be used to approximate such parameters as transistor I-V curves.
Such modeling is called Technology Computer-Aided Design (TCAD). The operation of a semiconductor device is described by a system of Partial Differential Equations (PDEs). The correct approximation (discretization) of PDEs, and, in particular, of semiconductor device equations (see [1]), is a basis of the numerical method. Modern devices might have anisotropy (i.e. the physical properties of materials are different for different directions). Among the reasons why devices have anisotropy can be the anisotropic properties of materials (such as Silicon Carbide) and/or the influence of mechanical stress which plays a very important role in the fabrication of modern devices. Anisotropy in a body to be modeled, at least for the types of anisotropy detailed herein, appear in the “transport coefficients” for the phenomenon being modeled. For example, in current continuity equations, the transport coefficient is an anisotropic mobility tensor. In the Poisson equation, the transport coefficient is an anisotropic dielectric permittivity tensor. In the heat equation, the transport coefficient is an anisotropic thermal conductivity tensor.
The main numerical method used in TCAD is the Finite Volume Method (see [2]) with Scharfetter-Gummel discretization (see [3]) of the semiconductor transport equations. The discretization is an approximation of the partial differential equations, and is often performed on Delaunay mesh (see, for example, [4]). The Scharfetter-Gummel approximation on Delaunay mesh has a few important properties; one of them guarantees that the numerically computed carrier density is always positive. Such a scheme is called monotone. However, the Scharfetter-Gummel approximation is monotone only for isotropic equations, and it may have non-physical negative solutions if directly applied to anisotropic equations.
There are limited cases when anisotropy can be easily taken into account in the Finite Volume method. In particular, if a simulation is performed using a tensor product mesh, and if the direction of anisotropy coincides with a mesh axis, then the discretization is simple; the situation is similar when the tensor product mesh is divided into triangles (tetrahedrons in 3D), but the direction of anisotropy still coincides with the edges of the right angle. However, these limitations are too serious: first, bodies having complicated geometry require a more general mesh which, for example, can be adjusted to non-rectilinear geometry; and second, in many practical applications, for example when the anisotropy is caused by stress engineering, the anisotropy is arbitrarily oriented and not known in advance.
This invention relates to the numerical discretization methods for simulating semiconductor devices, and in particular, to the construction of a new approximation for anisotropic semiconductor equations.